10.1515/msr-2015-0042
MEASUREMENT SCIENCE REVIEW, Volume 15, No. 6, 2015
Circular Samples as Objects for Magnetic Resonance Imaging Mathematical Simulation, Experimental Results
Ivan Frollo, Andrej Krafčík, Peter Andris, Jiří Přibil, Tomáš Dermek
Institute of Measurement Science, Slovak Academy of Sciences, Bratislava, Slovakia, frollo@savba.sk
Circular samples are the frequent objects of "in-vitro" investigation using imaging method based on magnetic resonance
principles. The goal of our investigation is imaging of thin planar layers without using the slide selection procedure, thus only 2D
imaging or imaging of selected layers of samples in circular vessels, eppendorf tubes,.. compulsorily using procedure "slide
selection". In spite of that the standard imaging methods was used, some specificity arise when mathematical modeling of these
procedure is introduced. In the paper several mathematical models were presented that were compared with real experimental
results. Circular magnetic samples were placed into the homogenous magnetic field of a low field imager based on nuclear
magnetic resonance. For experimental verification an MRI 0.178 Tesla ESAOTE Opera imager was used.
Keywords: Circular physical objects, mathematical modeling, magnetic susceptibility, Magnetic Resonance Imaging.
2. SUBJECT & METHODS
1. INTRODUCTION
I
MAGING METHODS used for biological and physical
structure investigation, based on Nuclear Magnetic
Resonance (NMR), have become a regular diagnostic
procedure. Specific occurrence is observed when an object,
consisting of a soft magnetic material, is inserted into a
static homogeneous magnetic field. This results in small
variations of the static homogeneous magnetic field near the
sample. The acquired image represents a modulation of the
basic homogeneous magnetic field of the imager detectable
by gradient-echo (GRE) or spin-echo (SE) imaging
sequences.
First attempts of a direct measurement of the magnetic
field variations created in living and physical tissues by a
simple wire fed by a current were published in [1]. A
method utilizing the divergence in gradient strength that
occurs in the vicinity of a thin current-carrying copper wire
was introduced in [2]. A simple experiment with thin, pulsed
electrical current-carrying wire and imaging of a magnetic
field, using a plastic sphere filled with agarose gel as
phantom, was published in [3]. Comparison of traditional
segmentation methods with 2D active contour methods was
discussed in [4]. Single biogenic soft magnetite nanoparticle
physical characteristics in biological objects were
introduced in [5]. Magnetic resonance imaging of the static
magnetic field distortion caused by magnetic nanoparticles:
Simulation and experimental verification were published in
[6].
In the paper several mathematical models are presented,
that were compared with real experimental results. Circular
magnetic samples were placed into the homogenous
magnetic field of a low field imager. Magnetic resonance
imaging method used for soft magnetic material detection,
computation of the magnetic field variations based on theory
of magnetism and comparison of theoretical results with
experimental images are the main goal of the next text. Via
experiments we want to prove that the proposed method is
perspective for soft magnetic materials testing using
magnetic resonance imaging methods.
In our experiments a soft ferromagnetic or paramagnetic
object was placed into the homogeneous magnetic field of
an MRI imager. The homogeneous magnetic field near the
sample is deformed. For simplicity and easy experimental
verification a circular object was selected and theoretically
analyzed.
A. Circular sample - theoretical principles
Let us suppose that the soft magnetic planar layer is
positioned in the x-y plane of the rectangular coordinate
system and the thickness of the layer is neglected.
According to Fig.1., we suppose the circular layers is
limited by diameter R, (R` for annulus). Let us suppose that
dS[x,y] is an elementary surface element. The static
magnetic field B0 of the MR imager is parallel with the zaxis. The task is to calculate the Bz component of the
magnetic field in the point A[x0,y0,z0].
Fig.1. Fundamental configuration of the circular planar magnetic
layer. The sample is positioned in x-y plane of the rectangular
coordinate system. The thickness of the layers is neglected.
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For theoretical analysis it is possible to consider the
magnetic double layer as magnetic dipoles continuously
distributed in the surface of the layer S. Then the overall
magnetic moment M relating to the surface S could be
expressed by a surface integral:
Direct evaluation of Eq. 3. is quite problematic. Some
problems can be solved analytically in other way, directly
from Maxwell equations. But still, almost everything is
solvable numerically, although approximatelly.
B. Calculation using analytical solution
ms =
∫
M S (r ) dS
and
MS = I n
According to Fig.2., we suppose three circular concentric
vessels. Relative permeabilities: μ1, μ2, environment
permeability: μ0.
(1)
S
where: ms - is magnetic dipole moment in a particular point,
r - is a position vector, I - is a current equivalent to planar
density of dipole moment of the magnetic double layer, and
n - is a unit normal vector of surface S in a particular point.
Magnetic double layer is considered having a
homogeneous density of the dipole moment that is oriented
in every point the direction of surface normal vector
perpendicular to the layer surface. Bearing in mind the
superposition principle it is possible to express the vector
potential of the double layer in the shape of surface integral:
A(r ) =
µ 0 I dS × r µ 0 I
1
=
grad  × dS
3
∫
∫
4π S r
4π S
r
Fig.2. Basic configuration of the magnetic planar layer circular
sample positioned in x-y plane of the rectangular coordinate
system.
(2)
After using general formula for vector potential and
magnetic induction B = curl A , we get the final formula
for magnetic induction of the magnetic double layer and
applicable also for the close current loop calculation as
follows:
The radial distance of a position vector in cylindrical
coordinate system can be expressed as:
µI
B ≡ curl A = 0
4π
Because of the symmetry of configuration, Laplace
equation can be expressed in cylindrical coordinate system
as
 3r (dS ⋅ r ) dS 
∫S  r 5 − r 3  ,
(3)
ρ ≡ x2 + y2 .
∇ 2 A( ρ , ϕ , z ) ≡
where the position vectors according to the Fig.1. can be
exppressed as:
r = r1 − r2
≡
and
r = ( x 0 − x ) 2 + ( y 0 − y) 2 + ( z 0 − z ) 2 .
B z ( x0 , y 0 , z 0 ) =
µ 0 I  3( z 0 − z ) 2 1 

− 3  dxdy
4π ∫S 
r5
r 
in Cartesian coordinate system, where r is in form (4).
∂ 2 A 1 ∂A 1 ∂ 2 A ∂ 2 A
+
+
+
= 0,
∂ρ 2 ρ ∂ρ ρ 2 ∂ϕ 2 ∂z 2
(7)
and solved analytically by separation of variables. For nonzero z-component of vector potential in k-th cylindrical layer
or environment we can write [10]:
(4)
D 

Ak =  C k ρ + k  cos ϕ ,
ρ 

If we consider single circular sample with normal vector
n = [0, 0,1] , equation (3) can be explicitly rewriten to the
form
µ I  3( x − x)( z0 − z ) 
B x ( x0 , y 0 , z 0 ) = 0 ∫  0
 dxdy
4π S 
r5

µ I  3( y0 − y )( z 0 − z ) 
B y ( x0 , y 0 , z 0 ) = 0 ∫ 
 dx dy
4π S 
r5

(6)
(8)
what yields radial and tangential components of magnetic
flux density in the form:
Bρk =
(5)

D
1 ∂Ak
= − C k + k2
ρ ∂ϕ
ρ


 sin ϕ ,


∂Ak
D
= − C k − k2
∂ρ
ρ


 cos ϕ ,

Bϕk = −
(9)
where Ck and Dk are constants calculated for every region
(domain). In material domain where ρ = 0 , vector potential
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has to be finite, what is fulfilled only if D
= 0 . In a
ρ =0
a)
similar way, for domain where ρ → ∞ ( µ = µ 0 ), magnetic
field has the intensity same as original external
homogeneous magnetic field, therefore C
= µ0 H 0 .
ρ →∞
Remaining constants can be determined from boundary
conditions on each cylindrical interface between two
material domains (e.g. I and II with radius RI), rewritten
explicitly in the form:
BρI
1
µI
BϕI
ρ = RI
ρ = RI
= BρII
=
1
µ II
ρ = RI
BϕII
,
(10)
ρ = RI
,
b)
1. Calculation example of 1 cylinder in H0
For one magnetic cylinder with axis identical with z-axis
of Cartesian coordinate system, whose radius is R, Fig.2.,
and permeability is μ1, located in surrounding environment
with permeability μ2 = μ0 ≡ 4π × 10−7 N/A2 and magnetic
field intensity H0 = [0, H0, 0], magnetic vector potential (8)
can be calculated for k-th medium domain directly by
solving system of linear equations, derived from boundary
conditions (10), expressed in matrix notation as:
Fig.3. Magnetic induction distribution of originally homogeneous
magnetic field H0 = μ0B0, B0 = 0.178 T affected by one magnetic
cylinder with permeability: μ1=2μ0 in environment with
permeability μ2 = μ0 ≡ 4π × 10−7 N/A2.
Solution of this system of linear equations is shown in
particular case for R = a, and μ1, μ2 = μ0 .
C1 =
2. Calculation example of 3 cylinders in H0
2 H 0 µ 0 µ1
µ0 + µ1
D1 = 0
For three concentric magnetic cylinders with axis identical
with z-axis of Cartesian coordinate system, whose radii
comply R1 < R2 < R3 and permeabilities are μ1, μ2, μ3
located in surrounding environment with permeability μ4
and magnetic field intensity H0 = [0, H0, 0], magnetic vector
potential (8) can be calculated for k-th medium domain
directly by solving system of linear equations, derived from
boundary conditions (10), expressed in matrix notation as
(11)
C2 = H 0 µ 0
D2 = −
H 0 a 2 µ 0 (µ0 − µ1 )
µ 0 + µ1
Calculation results are depicted in Fig.3. The pictures
illustrates the magnetic induction distribution of originally
static homogeneous magnetic field H0 = μ0B0, B0 = 0.178 T
affected by one magnetic cylinder. The environment is
declared with permeability μ2 = μ0. The range of calculated
values is: 0.12 - 0.24 Tesla. The values above the middle
value 0.178 T are typical in NMR images of circular
samples after application of GRE measuring sequence. The
values below the middle value are responsible for border
phenomena in the final images.
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MEASUREMENT SCIENCE REVIEW, Volume 15, No. 6, 2015
Solution of this system of linear equations is shown in
particular case, for:
R1 < R2 < R3, where R1 = a, R2 = b, R3 = c.
μ1 = μ0, μ2 = 2 μ0, μ3 = 3 μ0, μ4 = μ0,
and μ0 = 4π×10−7 N/A2 is vacuum permeability.
CI = −H 0
24 b 2 c 2 µ 0
,
5a 2 b 2 − 2a 2 c 2 + 3b 4 − 30b 2 c 2
DI = 0,
36 b 2 c 2 µ 0
,
5a 2 b 2 − 2a 2 c 2 + 3b 4 − 30b 2 c 2
12a 2 b 2 c 2 µ 0
DII = H 0 2 2
,
5a b − 2a 2 c 2 + 3b 4 − 30b 2 c 2
3c 2 µ 0 (a 2 + 15b 2 )
C III = − H 0 2 2
,
5a b − 2a 2 c 2 + 3b 4 − 30b 2 c 2
3c 2 µ 0 (5a 2 b 2 + 3b 4 )
DIII = H 0 2 2
,
5a b − 2a 2 c 2 + 3b 4 − 30b 2 c 2
C IV = µ 0 H 0 ,
C II = − H 0
DIV = H 0
Fig.4. Mathematical model of the distribution of homogeneous
magnetic field H0, lines of force affected by 2 concentric cylinders
with relative permeabilities: μ1=2 and μ2=3, μ0 is an environment
permeability.
(12)
a)
c 2 µ 0 (10a 2 b 2 − a 2 c 2 + 6b 4 − 15b 2 c 2 )
.
5a 2 b 2 − 2a 2 c 2 + 3b 4 − 30b 2 c 2
Resultant z-component of vector potential respecting four
regions using constants CI to CIV and DI to DIV in planar
approximation is as follows:
 AI ,
A ,
 II
A=
 AIII ,
 AIV ,
ρ ≤ a,
for a < ρ ≤ b,
for b < ρ ≤ c,
for c < ρ ,
for
b)
(13)
and vector potential A=[0,0,A]. Resultant flux density of
magnetic field will be:
B = ∇×A.
(14)
Its magnetic flux density lines is possible to draw as a
stream plot, relative values:
StreamPlot[{Hr}, {x,-0.08, 0.08}, {y,- 0.08, 0.08}] .
Mathematical model of the distribution of homogeneous
magnetic field H0, lines of force affected by 3 concentric
cylinders is depicted in Fig 4.
Calculation results of magnetic induction distribution of
stationary homogeneous magnetic field H0 affected by 3
concentric magnetic cylinders are depicted in Fig.5. These
are theoretical examples using relative values of magnetic
permeabilities. The range of calculated values: 0.15 - 0.29 T.
The values above and below the middle value 0.178 T are
typical in NMR images of circular samples after application
of GRE measuring sequence. The permeability values are
relative. For simulation of real samples it is necessary to
use the actual values.
Fig.5. Magnetic induction distribution of static homogeneous
magnetic field H0 affected by 3 concentric magnetic cylinders with
permeabilities: μ1 = μ0, μ2 = 2 μ0, μ3 = 3μ0 in environment with
permeability μ4 = μ0.
C. Experimental results
For experimental evaluation of the mathematical modeling
we have chosen two simple laboratory arrangements for
application of magnetic resonance imaging methods.
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1. Liquid circular samples.
Cylinder vessels placed in a rectangular plastic holder
filled with the tap water were used. Cylinder vessels were
filled with distilled water doped by several drops of
magnetic liquid based on Dextran (polysaccharide).
For experimental evaluation of the mathematical
modeling, two samples - circular vessels filled with doped
water were placed to the centre of the holder. The central
vessel was filled with tap water. Diameters of the vessels:
D1=18, D2=40, D3=56 mm, wall thickness: 1 mm.
Resultant NMR image using GRE imaging sequence is
depicted in Fig.6.
An NMR image of the magnetic annulus (cut from the data
diskette) is depicted in Fig.8. The annulus was placed to the
plastic holder and flooded by the water. The image
represents magnetic field variations detected via water
medium. The measurement was performed two times with
different echo times TE. The Fourier transformed data
matrices were ratio processed to prevent phase wrapping
and to remove some distortions [14]. The magnetic field
scale relates to the circular image of the sample.
Fig.8. Left: Plastic holder used for sample positioning. Dish for the
testing liquid. Circular sample dimensions: D1=60 mm, D2=50 mm.
Right: Magnitude image of the magnetic annulus. Size of the
measured matrix: 256 x 256.
Fig.6. NMR image of liquid samples (three circular glass vessels
filled with doped water in the rectangular boundary vessel - holder
filled with tap water) using GRE imaging sequence: TR = 440 ms,
TE = 10 ms. Thickness of the imaged layer: 2 mm.
Similar imaging sequences were used for imaging of
commercial metal coins, see Fig.9. In the MR image an
interesting structure is visible that illustrates the metal
inhomogeneities inside the coins.
2. Weak magnetit solid circular samples.
For sample positioning a plastic holder was constructed. The
principal experimental arrangement is depicted in Fig.7.
Fig.9. Examples of imaging of commercial coins. Left: Photo of 7
coins. Right: MR image, material inhomogeneities are clearly
visible when GRE sequence was used.
Fig.7. Sample positioning in the plastic holder. Static magnetic
field of the imager B0 is perpendicular and RF field is parallel with
the plane of the holder.
A radio frequency transducing coil (solenoid) together
with the plastic was placed into the centre of the permanent
magnet, perpendicular to the magnetic field (B0) orientation.
The “gradient-echo” NMR sequences were selected for the
measurements [8], [9]. A special feature of this sequence is
its sensitivity to basic magnetic field inhomogeneities. The
actual parameters used in presented experiments are as
follows: RF pulse: 900, repetition time: TR = 800 ms, echo
time: TE=10 ms, number of averages: 6, max. measured
matrix: 256 x 238, field of view: 140 x 130.
For experimental verification the MRI 0.178 Tesla
ESAOTE Opera imager (Esaote, Genoa, Italy) with vertical
orientation of the basic magnetic field was used.
Several gradient-echo sequences were tested. It is evident,
that every designed sequence is generating different image.
We tried to find the best correlation between calculated
results and resultant images.
3. CONCLUSIONS
The objective of this paper was to present a mathematical
simulation of soft magnetic circular objects properties and
using experimental magnetic resonance imaging methods
for simple circular objects - vessels filled with doped water
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MEASUREMENT SCIENCE REVIEW, Volume 15, No. 6, 2015
by very diluted magnetic liquids and magnetit solid circular
samples using special plastic holder for sample positioning.
Mathematical analysis of circular objects, representing a
shaped magnetic soft layer, showed theoretical possibilities
to calculate magnetic field around any type of sample. Our
mathematical models proved that it is possible to map the
magnetic field variations - line of force - and to image the
specific structures of selected samples placed into a special
plastic holders. The calculations were performed with
relative values of input quantities, permeabilities and
dimensions. The mathematical model was described in the
form of general formulas.
For experimental presentation a classical gradient-echo
measuring sequences were used.
The resultant MR images are encircled by narrow stripes
that optically extend the width of the sample. This
phenomenon is typical for susceptibility imaging, when one
needs to measure local magnetic field variations
representing sample properties [11], [12], [13].
All experimental results are in relative good correlation
with the mathematical simulations in spite of used
comparative quantities. The proposed methods confirm the
possible suitability of the proposed method for detection of
weak magnetic materials in liquid or thin layer low magnetic
objects using the MRI methods.
[4]
[5]
[6]
[7]
[8]
[9]
ACKNOWLEDGMENT
This work was supported by the Slovak Scientific Grant
Agency VEGA 2/0013/14, within the project of the Slovak
Research and Development Agency Nr. APVV-0431-12 and
project of the University scientific park for biomedicine,
Bratislava, ITMS: 26240220087.
[10]
[11]
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Received September 11, 2015.
Accepted December 2, 2015.